Optimal. Leaf size=69 \[ \frac {3}{2} x^2 \text {Li}_2\left (-e^x\right )-\frac {3}{2} x^2 \text {Li}_2\left (e^x\right )-3 x \text {Li}_3\left (-e^x\right )+3 x \text {Li}_3\left (e^x\right )+3 \text {Li}_4\left (-e^x\right )-3 \text {Li}_4\left (e^x\right )+x^3 \tanh ^{-1}\left (e^x\right ) \]
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Rubi [A] time = 0.12, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {2249, 206, 2245, 6213, 2531, 6609, 2282, 6589} \[ \frac {3}{2} x^2 \text {PolyLog}\left (2,-e^x\right )-\frac {3}{2} x^2 \text {PolyLog}\left (2,e^x\right )-3 x \text {PolyLog}\left (3,-e^x\right )+3 x \text {PolyLog}\left (3,e^x\right )+3 \text {PolyLog}\left (4,-e^x\right )-3 \text {PolyLog}\left (4,e^x\right )+x^3 \tanh ^{-1}\left (e^x\right ) \]
Antiderivative was successfully verified.
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Rule 206
Rule 2245
Rule 2249
Rule 2282
Rule 2531
Rule 6213
Rule 6589
Rule 6609
Rubi steps
\begin {align*} \int \frac {e^x x^3}{1-e^{2 x}} \, dx &=x^3 \tanh ^{-1}\left (e^x\right )-3 \int x^2 \tanh ^{-1}\left (e^x\right ) \, dx\\ &=x^3 \tanh ^{-1}\left (e^x\right )+\frac {3}{2} \int x^2 \log \left (1-e^x\right ) \, dx-\frac {3}{2} \int x^2 \log \left (1+e^x\right ) \, dx\\ &=x^3 \tanh ^{-1}\left (e^x\right )+\frac {3}{2} x^2 \text {Li}_2\left (-e^x\right )-\frac {3}{2} x^2 \text {Li}_2\left (e^x\right )-3 \int x \text {Li}_2\left (-e^x\right ) \, dx+3 \int x \text {Li}_2\left (e^x\right ) \, dx\\ &=x^3 \tanh ^{-1}\left (e^x\right )+\frac {3}{2} x^2 \text {Li}_2\left (-e^x\right )-\frac {3}{2} x^2 \text {Li}_2\left (e^x\right )-3 x \text {Li}_3\left (-e^x\right )+3 x \text {Li}_3\left (e^x\right )+3 \int \text {Li}_3\left (-e^x\right ) \, dx-3 \int \text {Li}_3\left (e^x\right ) \, dx\\ &=x^3 \tanh ^{-1}\left (e^x\right )+\frac {3}{2} x^2 \text {Li}_2\left (-e^x\right )-\frac {3}{2} x^2 \text {Li}_2\left (e^x\right )-3 x \text {Li}_3\left (-e^x\right )+3 x \text {Li}_3\left (e^x\right )+3 \operatorname {Subst}\left (\int \frac {\text {Li}_3(-x)}{x} \, dx,x,e^x\right )-3 \operatorname {Subst}\left (\int \frac {\text {Li}_3(x)}{x} \, dx,x,e^x\right )\\ &=x^3 \tanh ^{-1}\left (e^x\right )+\frac {3}{2} x^2 \text {Li}_2\left (-e^x\right )-\frac {3}{2} x^2 \text {Li}_2\left (e^x\right )-3 x \text {Li}_3\left (-e^x\right )+3 x \text {Li}_3\left (e^x\right )+3 \text {Li}_4\left (-e^x\right )-3 \text {Li}_4\left (e^x\right )\\ \end {align*}
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Mathematica [A] time = 0.05, size = 89, normalized size = 1.29 \[ \frac {3}{2} x^2 \text {Li}_2\left (-e^x\right )-\frac {3}{2} x^2 \text {Li}_2\left (e^x\right )-3 x \text {Li}_3\left (-e^x\right )+3 x \text {Li}_3\left (e^x\right )+3 \text {Li}_4\left (-e^x\right )-3 \text {Li}_4\left (e^x\right )-\frac {1}{2} x^3 \log \left (1-e^x\right )+\frac {1}{2} x^3 \log \left (e^x+1\right ) \]
Antiderivative was successfully verified.
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fricas [C] time = 0.45, size = 71, normalized size = 1.03 \[ \frac {1}{2} \, x^{3} \log \left (e^{x} + 1\right ) - \frac {1}{2} \, x^{3} \log \left (-e^{x} + 1\right ) + \frac {3}{2} \, x^{2} {\rm Li}_2\left (-e^{x}\right ) - \frac {3}{2} \, x^{2} {\rm Li}_2\left (e^{x}\right ) - 3 \, x {\rm polylog}\left (3, -e^{x}\right ) + 3 \, x {\rm polylog}\left (3, e^{x}\right ) + 3 \, {\rm polylog}\left (4, -e^{x}\right ) - 3 \, {\rm polylog}\left (4, e^{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int -\frac {x^{3} e^{x}}{e^{\left (2 \, x\right )} - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 74, normalized size = 1.07 \[ -\frac {x^{3} \ln \left (-{\mathrm e}^{x}+1\right )}{2}+\frac {x^{3} \ln \left ({\mathrm e}^{x}+1\right )}{2}+\frac {3 x^{2} \polylog \left (2, -{\mathrm e}^{x}\right )}{2}-\frac {3 x^{2} \polylog \left (2, {\mathrm e}^{x}\right )}{2}-3 x \polylog \left (3, -{\mathrm e}^{x}\right )+3 x \polylog \left (3, {\mathrm e}^{x}\right )+3 \polylog \left (4, -{\mathrm e}^{x}\right )-3 \polylog \left (4, {\mathrm e}^{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.47, size = 71, normalized size = 1.03 \[ \frac {1}{2} \, x^{3} \log \left (e^{x} + 1\right ) - \frac {1}{2} \, x^{3} \log \left (-e^{x} + 1\right ) + \frac {3}{2} \, x^{2} {\rm Li}_2\left (-e^{x}\right ) - \frac {3}{2} \, x^{2} {\rm Li}_2\left (e^{x}\right ) - 3 \, x {\rm Li}_{3}(-e^{x}) + 3 \, x {\rm Li}_{3}(e^{x}) + 3 \, {\rm Li}_{4}(-e^{x}) - 3 \, {\rm Li}_{4}(e^{x}) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ -\int \frac {x^3\,{\mathrm {e}}^x}{{\mathrm {e}}^{2\,x}-1} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ - \int \frac {x^{3} e^{x}}{e^{2 x} - 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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